In recent years, active research has been conducted on nanoscale electronic devices, quantum computers, and the like. In the case where an electronic device is a structure of a scale equal to or smaller than about 10 nm which is a mean free path of electrons, quantum effects become noticeable in electrical and magnetic properties such as electrical conductance. Thus, it is difficult to design and develop these nanoscale electronic devices and quantum computers according to an ordinary method that is based on macroscale electrical and magnetic properties. Accordingly, a numerical simulation apparatus capable of performing a numerical simulation of dynamic behavior of electrons such as scattering, transmission, reflection, interference, oscillation, attenuation, and excitation with high speed and precision is needed for the design and development of these nanoscale electronic devices and quantum computers.
For instance, with miniaturization and high integration of a nanoscale electronic device, a conduction electron path becomes an atomic size (for example, see (a) in FIG. 1). As a result, electronic transport properties specific to a nanoregion, such as ballistic conduction, appear. Therefore, in evaluations of leakage currents of silicon oxide and conduction properties of a quantum wire (for example, see (b) in FIG. 1), it is necessary to numerically simulate conduction properties using the time dependent Schrödinger equation.
In a numerical simulation of conduction properties, the steady-state Schrödinger equation for a scattering problem is typically used. This being so, various methods for numerically simulating the steady-state Schrödinger equation for a scattering problem have been proposed. One of them is the Overbridging Boundary-Matching (OBM) method which employs the real-space finite-difference method (for example, see Non-patent Reference 1).
On the other hand, there is also a method for clarifying dynamic behavior of electrons by directly solving the time dependent Schrödinger equation. In this case, the time dependent Schrödinger equation defined by the following Expression (1) is used.
                    [                  Expression          ⁢                                          ⁢          1                ]                                                                      ⅈ          ⁢                      ∂                          ∂              t                                ⁢                      Ψ            ⁡                          (                              r                ,                t                            )                                      =                  H          ⁢                                          ⁢                      Ψ            ⁡                          (                              r                ,                t                            )                                                          (        1        )            
H is a Hamiltonian, and is given by the following Expression (2). Δ is a Laplacian, and is given by the following Expression (3). V(r, t) is a potential of an electron. Atomic units are used as a system of units. An electron mass is m=1, an elementary charge is e=1, and h/2n=1, where h is a Planck's constant.
                    [                  Expression          ⁢                                          ⁢          2                ]                                                            H        =                                            -                              1                2                                      ⁢            Δ                    +                      V            ⁡                          (                              r                ,                t                            )                                                          (        2        )                                [                  Expression          ⁢                                          ⁢          3                ]                                                            Δ        =                                            ∂              2                                      ∂                              x                2                                              +                                    ∂              2                                      ∂                              y                2                                              +                                    ∂              2                                      ∂                              z                2                                                                        (        3        )            
Furthermore, expressing a solution of the above Expression (1) by a path integral using the propagator K (Feynman Kernel) leads to the following Expression (4), where Ψ(r, t0) is a wave function at initial time t0.
                    [                  Expression          ⁢                                          ⁢          4                ]                                                                      Ψ          ⁡                      (                          r              ,              t                        )                          =                              ∫                          -              ∞                        ∞                    ⁢                                    K              ⁡                              (                                  r                  ,                                      t                    ;                                          r                      ′                                                        ,                                      t                    0                                                  )                                      ⁢                          Ψ              ⁡                              (                                                      r                    ′                                    ,                                      t                    0                                                  )                                      ⁢                                                  ⁢                                          ⅆ                3                            ⁢                              r                ′                                                                        (        4        )            
It is known that the above Expression (4) can be expressed by a path integral. For instance, the wave function after short time period Δt is defined by the following Expression (5) (for example, see Non-patent Reference 2).
                    [                  Expression          ⁢                                          ⁢          5                ]                                                                      Ψ          ⁡                      (                          r              ,                              t                +                                  Δ                  ⁢                                                                          ⁢                  t                                                      )                          =                                            1                              2                ⁢                π                ⁢                                                                  ⁢                ⅈ                ⁢                                                                  ⁢                Δ                ⁢                                                                  ⁢                t                                              ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          ⅇ                                  ⅈ                  ⁢                                                                                    (                                                  r                          -                                                      r                            ′                                                                          )                                            2                                                              2                      ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                                                                  ⁢                              ⅇ                                                      -                    ⅈΔ                                    ⁢                                                                          ⁢                                      tV                    ⁡                                          (                                                                                                    r                            +                                                          r                              ′                                                                                2                                                ,                        t                                            )                                                                                  ⁢                                                          ⁢                              Ψ                ⁡                                  (                                                            r                      ′                                        ,                    t                                    )                                            ⁢                                                ⅆ                  3                                ⁢                                  r                  ′                                                                                        (        5        )                Non-patent Reference 1: K. Hirose, T. Ono et al., First-Principles Calculations in Real-Space Formalism, Imperial College Press, London (2005)    Non-patent Reference 2: R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, translated by Kazuo Kitahara, McGraw-Hill (July 1990)